Solving Quadratic Inequalities: x² + 9x + 14 > 0, r²

Math Problem Statement

Find the solution set of each of the following quadratic inequalities: 1. x² + 9x + 14 > 0 2. r² - 10r + 16 < 0 3. x² + 6x ≥ -5 4. m² - 7m ≤ 10

Solution

The image displays the following set of quadratic inequalities to solve:

$x^2 + 9x + 14 > 0$
$r^2 - 10r + 16 < 0$
$x^2 + 6x \geq -5$
$m^2 - 7m \leq 10$

Let's go through these one by one and solve them.

1. $x^2 + 9x + 14 > 0$

First, solve the corresponding equation $x^2 + 9x + 14 = 0$ by using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ For this equation, $a = 1$ , $b = 9$ , and $c = 14$ . Substituting these values:

= \frac{-9 \pm \sqrt{81 - 56}}{2} = \frac{-9 \pm \sqrt{25}}{2} = \frac{-9 \pm 5}{2}$$ So, $$x_1 = \frac{-9 + 5}{2} = -2$$ and $$x_2 = \frac{-9 - 5}{2} = -7$$. The quadratic equation has roots $$x = -2$$ and $$x = -7$$. Since we are solving $$x^2 + 9x + 14 > 0$$, the inequality will be positive for values outside the interval $$(-7, -2)$$. Thus, the solution is: $$x \in (-\infty, -7) \cup (-2, \infty)$$ ### 2. $$r^2 - 10r + 16 < 0$$ First, solve the quadratic equation $$r^2 - 10r + 16 = 0$$: $$r = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(1)(16)}}{2(1)} = \frac{10 \pm \sqrt{100 - 64}}{2} = \frac{10 \pm \sqrt{36}}{2} = \frac{10 \pm 6}{2}$$ Thus, $$r_1 = \frac{10 + 6}{2} = 8$$ and $$r_2 = \frac{10 - 6}{2} = 2$$. Since we are solving $$r^2 - 10r + 16 < 0$$, the inequality will be negative between the roots. Therefore, the solution is: $$r \in (2, 8)$$ ### 3. $$x^2 + 6x \geq -5$$ Rewrite the inequality: $$x^2 + 6x + 5 \geq 0$$ Solve the equation $$x^2 + 6x + 5 = 0$$ using the quadratic formula: $$x = \frac{-6 \pm \sqrt{6^2 - 4(1)(5)}}{2(1)} = \frac{-6 \pm \sqrt{36 - 20}}{2} = \frac{-6 \pm \sqrt{16}}{2} = \frac{-6 \pm 4}{2}$$ Thus, $$x_1 = \frac{-6 + 4}{2} = -1$$ and $$x_2 = \frac{-6 - 4}{2} = -5$$. The quadratic expression is greater than or equal to zero outside the interval $$(-5, -1)$$, and equal to zero at $$x = -5$$ and $$x = -1$$. Therefore, the solution is: $$x \in (-\infty, -5] \cup [-1, \infty)$$ ### 4. $$m^2 - 7m \leq 10$$ Rewrite the inequality: $$m^2 - 7m - 10 \leq 0$$ Solve the quadratic equation $$m^2 - 7m - 10 = 0$$ using the quadratic formula: $$m = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(1)(-10)}}{2(1)} = \frac{7 \pm \sqrt{49 + 40}}{2} = \frac{7 \pm \sqrt{89}}{2}$$ The roots are $$m = \frac{7 \pm \sqrt{89}}{2}$$, which are approximately $$m_1 \approx 8.22$$ and $$m_2 \approx -1.22$$. Since we are solving $$m^2 - 7m - 10 \leq 0$$, the inequality will be satisfied between the roots. Therefore, the solution is approximately: $$m \in [-1.22, 8.22]$$ Would you like more details or any specific graphing of these inequalities? Here are five related questions: 1. How do we know the sign of a quadratic function changes at its roots? 2. How can the quadratic formula help when solving inequalities? 3. What is the difference between solving quadratic inequalities and equations? 4. How would you graph the solution for each inequality on a number line? 5. How do we handle cases where a quadratic inequality includes "greater than or equal to"? **Tip:** Always check if the quadratic can be factored easily before using the quadratic formula—it can save time!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Quadratic Inequalities
Solving Quadratics
Algebra

Formulas

Quadratic Formula: x = (-b ± √(b² - 4ac)) / 2a
Factoring Quadratics

Theorems

Quadratic formula
Sign of Quadratic Functions Based on Roots

Suitable Grade Level

Grades 9-12

Related Recommendation

Solving Quadratic Inequalities for Various Cases

Solving Quadratic Inequalities: x² + 9x + 14 > 0 and x² – 5x - 14 ≥ 0

Solving Quadratic Inequalities with Factoring and Graphical Analysis

Solving Quadratic Inequalities with Examples: X^2 - 9x + 8 ≥ 0 and More

Solve Quadratic and Rational Inequalities Involving Factoring